I think you have that backwards dude.
The CPU cores always have a higher temp then the heatspreader.
At idle the main temp for my quad (which would be the heatspreader temp) is 25C. Each core at idle runs at 28C.
I use Gigabyte's EasyTune5 Pro to check my cpu temp, and AMD OverDrive Utility to see the core temps. Everest is good too. It gives me the same readings.

Maybe you have a CPU that the heatspreader wasn't applied to correctly when manufactured?

Well don't the cores dissipate their hate to the spreader itself and that is the part that is hotter ? In all my readings throughout the years I noticed the spreader being hotter. My Athlon 64 x2 5600+ stock, with both cores at 100%, temp of the heat spreader can be in the 65-66C, while the CPU cores are 50-55C. and yes heatsink is applied correctly - think it's AMD that's cutting corners lately and their stock coolers are garbage, even before applying them I noticed the inconsistencies.